11 THE SPECTRAL THEORY OF HILBERT 407 13. Let E be a separable Hilbert space, and (Hn) a sequence of self-adjoint operators with the following property: there exists an interval [—M, M] in R such that, if P(X) = a0 H- atX + • • • 4- anXn is any polynomial with real coefficients and P(f) ^ 0 for — M ^ £ :g M, then also aQl + a^Hi -\-------h anHn ^ 0 (which implies inter alia that — M • I <; Hn <; M • I for all n). Show that there exists a separable Hilbert space G which is the Hilbert sum of E and a Hilbert space F, and a self-adjoint operator H on G such that Hn = PH" \ E for all n ^ 1, where P is the orthogonal pro- jection of G on E. (Using Problem 5 of Section 13.20, prove that for each pair (x, y) of elements of E there exists a real measure mXt y on [—M, M] such that (Hn - x | y) = J £" dmXt y(£\ (x \ y) = J dmXt y(g) for all n ^ 1.) Deduce that H2n + l == li-^nll^Gn + a • 14. Let E be a separable Hilbert space and T a continuous operator on E such that \\T\\ <£ 1. Then there exists a separable Hilbert space H, the Hilbert sum of E and a Hilbert space F, and a unitary operator U on H such that, if P is the orthogonal projection of H on E, then Tn = PU" | E for all /i S> 1. (Apply Problem 6 of Section 15.9, taking F = Z, the involution on F being n\— > — n, and the representation of T in H such that U(n) = J"1 for all n ;> 0. Observe that, for all ^ e E and all £ e C such that |f | < 1, we have and note that every # e E can be written in the form (I — t>T) - y for some y e E. + 00 In particular, this is so for every linear combination x — £ xne~ni<p, where the xn e E — 00 are zero except for finitely many indices n, and where £ = ret<p with r>0. Then express 2(rlB-m| «/(«- m) • m, n in terms of the left-hand side of (*), and let r tend to 1.) 00 Deduce that, if ]£ cn z" is a power series which converges absolutely on the circle |z| = l, and if u(T) = ^cnT", then the relation \u(z) \ ^ 1 (resp. &u(z)^Q) for | z\ g 1 implies that ||«(T)II g 1 (resp. u(T) + w(r*) ^ 0). (Note that «(T) = Pw(i7) | E.) 15. If N is a normal continuous operator on a separable Hilbert space E, and K a compact subset of C containing Sp(7V~), then we have a (nonfaithful) representation /W/(AO of ^c(K) in E ; the measures mXt y can be considered as measures on K satisfying (15.11.2) and (15.11.2.1). In particular, if U is a unitary operator on E, then/W/(C7) is a representation of ^C(U) in E. Let (en) be a Hilbert basis of E indexed by Z, and let U be the unitary operator on E such that U • en — en + 1. Show that the representation /t-*/(C7) of ^C(U) on E is topologically cyclic and that e0 is a totalizing vector; and that ^ = meo, e0 is the normalized Haar measure on U (cf. (7.4.2)), Deduce that Sp(CT) is the whole of the circle U, and also give a direct proof of this fact. Give examples of closed vector subspaces of E which are stable under U but not under £7*- U'1.and all